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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2011, Article ID 520273, 11 pagesdoi:10.1155/2011/520273

Research ArticleDistribution of Maps with Transversal HomoclinicOrbits in a Continuous Map Space

Qiuju Xing and Yuming Shi

Department of Mathematics, Shandong University, Jinan, Shandong 250100, China

Correspondence should be addressed to Yuming Shi, [email protected]

Received 19 January 2011; Accepted 25 April 2011

Academic Editor: Agacik Zafer

Copyright q 2011 Q. Xing and Y. Shi. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper is concerned with distribution of maps with transversal homoclinic orbits in a continu-ousmap space, which consists of continuousmaps defined in a closed and bounded set of a Banachspace. By the transversal homoclinic theorem, it is shown that the map space contains a dense setof maps that have transversal homoclinic orbits and are chaotic in the sense of both Li-Yorke andDevaney with positive topological entropy.

1. Introduction

Distribution of a set of maps with some dynamical properties in some continuous map spaceis a very interesting topic. In the 1960s, Smale [1] studied density of hyperbolicity. Somescholars believed that hyperbolic systems are dense in spaces of all dimensions, but it wasshown that the conjecture is false in the late 1960s for diffeomorphisms on manifolds ofdimension ≥2. The problem whether hyperbolic systems are dense in the one-dimension casewas studied by many scholars. It was solved in the C1 topology by Jakobson [2], a partialsolution was given in the C2 topology by Blokh and Misiurewicz [3], and C2 density wasfinally proved by Shen [4]. In 2007, Kozlovki et al. got the result in Ck topology; that is,hyperbolic (i.e., Axiom A) maps are dense in the space of Ck maps defined in a compactinterval or circle, k = 1, 2, . . . ,∞, ω [5]. At the same time, some other scholars consideredthe distribution of hyperbolic diffeomorphisms in Diff(M), where M is a manifold. Justlike the work of Smale, Palis [6, 7] gave the following conjecture: (1) any f ∈ Diff(M)can be approximated by a hyperbolic diffeomorphism or by a diffeomorphism exhibiting ahomoclinic bifurcation (tangency or cycle), (2) any diffeomorphism can be Cr approximatedby a Morse-Smale one or by one exhibiting transversal homoclinic orbit. Later, it was shownthat the conjecture (1) holds for C1 diffeomorphisms of surfaces [8]. And some good results

2 Abstract and Applied Analysis

have been obtained, such as any diffeomorphism can be C1 approximated by a Morse-Smaleone or by one displaying a transversal homoclinic orbit [9], any diffeomorphism can be C1

approximated by one that exhibits either a homoclinic tangency or a heterodimensional cycleor by one that is essentially hyperbolic [10].

In 1963, Smale gave thewell-known Smale-Birkhoff homoclinic theorem for diffeomor-phisms [11], from which one can easily know that if a diffeomorphism F on a manifold M

has a transversal homoclinic orbit, then there exists an integer k > 0 such that Fk is chaotic inthe sense of both Li-Yorke and Devaney. Later, in 1986, Hale and Lin introduced a generalizeddefinition of transversal homoclinic orbit for continuous maps and got the generalized Smale-Birkhoff homoclinic theorem, that is, a transversal homoclinic orbit implies chaos in the senseof both Li-Yorke and Devaney for continuous maps in Banach spaces [12]. In the meanwhile,some scholars studied the density of maps which are chaotic in the sense of Li-Yorke orDevaney. In particular, some results have been obtained in one-dimensional maps (cf. [13–15]).

Since 2004, Shi, Chen, and Yu extended the result about turbulent maps for one-dimensional maps introduced by Block andCoppel in 1992 [16] to maps inmetric spaces. Thismap is termed by a new terminology: coupled-expanding map. Under certain conditions,the authors showed that a strictly coupled-expanding map is chaotic [17, 18]. Applyingthis coupled-expansion theory, they extended the criterion of chaos induced by snap-backrepellers for finite-dimensional maps, introduced by Marotto in 1978 [19], to maps in metricspaces [17, 20, 21]. Recently, we studied the distribution of chaotic maps in continuous mapspaces, in which maps are defined in general Banach spaces and finite-dimensional normedspaces, and obtained that the following several types of chaotic maps are dense in some con-tinuous map spaces: (1)maps that are chaotic in the sense of both Li-Yorke and Devaney; (2)maps that are strictly coupled-expanding; (3) maps that have nondegenerated and regularsnap-back repeller; (4) maps that have nondegenerate and regular homoclinic orbit to arepeller [22].

In the present paper, we will construct a set of continuous chaotic maps with gener-alized transversal homoclinic orbits, and show that the set is dense in the continuous mapspace. The method used in the present paper is motivated by the idea in [22].

This paper is organized as follows. In Section 2, we first introduce some notationsand basic concepts including the Li-Yorke and Devaney chaos, hyperbolic fixed point, andtransversal homoclinic orbit, and then give a useful lemma. In Section 3, we pay our attentionto distribution of maps with generalized transversal homoclinic orbits in a continuous mapspace, in which every map is defined in a closed, bounded, and convex set or a closedbounded set in a general Banach space. Constructing a continuous map with a generalizedtransversal homoclinic orbit, we simultaneously show the density of maps which are chaoticin the sense of both Li-Yorke and Devaney in the map space.

2. Preliminaries

In this section, some notations and basic concepts are first introduced, including Li-Yorke andDevaney chaos, hyperbolic fixed point, and transversal homoclinic orbit. And then a usefullemma is given.

First, we give two definitions of chaos which will be used in the paper.

Abstract and Applied Analysis 3

Definition 2.1 (see [23]). Let (X, d) be a metric space, f : X → X a map, and S a set of X withat least two points. Then, S is called a scrambled set of f if, for any two distinct points x, y ∈ S,

lim infn→∞

d(fn(x), fn(y

))= 0, lim sup

n→∞d(fn(x), fn(y

))> 0. (2.1)

The map f is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scram-bled set S of f .

Definition 2.2 (see [24]). Let (X, d) be a metric space. A map f : V ⊂ X → V is said to be cha-otic on V in the sense of Devaney if

(i) the periodic points of f in V are dense in V ;

(ii) f is topologically transitive in V ;

(iii) f has sensitive dependence on initial conditions in V .

Now, we give the definition of hyperbolic fixed point.

Definition 2.3 (see [25]). Let X be a Banach space, U ⊂ X be a set, and F : U ⊂ X → X bea map. Assume that z is a fixed point of F, F is continuously differentiable in some neigh-borhood of z, and denote DF(z) the Frechet derivative of F at z. The fixed point z is calledhyperbolic if σ(DF(z)) ∩ {λ ∈ C : |λ| = 1} = ∅, where σ(A) denotes the spectrum of a linearoperator A. The hyperbolic fixed point z is called a saddle point if σ(DF(z)) ∩ {λ ∈ C : |λ| >1}/= ∅ and σ(DF(z)) ∩ {λ ∈ C : |λ| < 1}/= ∅.

If F is invertible, then for any p0 ∈ X, the setO+F(p0) := {Fk(p0) : k ≥ 0} is said to be the

forward orbit of F from p0, the set O−F(p0) := {Fk(p0) : k ≤ 0} is said to be the backward orbit

of F from p0. Since it is not required that F is invertible in this paper, a backward orbit of p0 isa set O−

F(p0) = {pj : j ≤ 0} with pj+1 = F(pj), j ≤ −1, which may not exist, or exist but may notbe unique. A whole orbit of p0 is the union O+

F(p0) ∪ O−F(p0), denoted by OF(p0) in the case

that it has a backward orbit O−F(p0). The stable set W

s(z, F) and the unstable set Wu(z, F) ofa hyperbolic fixed point z of F are defined by

Ws(z, F) :={p0 ∈ X: the forward orbit

{pj}+∞j=0 of F from p0 such that,

pj −→ z as j −→ +∞},

Wu(z, F) :={p0 ∈ X: there exists a backward orbit

{p−j

}+∞j=0 of F

from p0 such that p−j −→ z as j −→ +∞},

(2.2)

respectively. The local stable and unstable sets are defined by

Wsloc(z,U, F) := Ws(z, F) ∩U, Wu

loc(z,U, F) := Wu(z, F) ∩U, (2.3)

respectively, where U is some neighborhood of z. If U = B(z, r) or B(z, r) for some r > 0,then the corresponding local stable and unstable sets of F are denoted by Ws

loc(z, r, F)and Wu

loc(z, r, F), respectively. By the Stable Manifold Theorem [26], if F is continuously


differentiable in some neighborhood of a saddle point z, then there exists a neighborhoodU of z such that the corresponding local stable and unstable set of z are submanifolds of X,respectively.

In the following, we first give the definition that two manifolds intersect transversallyand then give the definition of transversal homoclinic orbit for continuous maps.

Definition 2.4 (see [25]). Two submanifolds V and W in a manifold M are transverse (in M)provided for any point q ∈ V ∩ W , we have that TqV + TqW = TqM, where TqV and TqWdenote the tangent spaces of V and W at q, respectively, and “+” means the sum of the twosubspaces (this allows for the possibility that V ∩W = ∅).

Remark 2.5. If M = Rn, then V and W in M are transverse (in M) provided for any pointq ∈ V ∩ W , we have that TqV + TqW = Rn. Obviously, if dim TqV + dim TqW = n, then thesum of the two subspaces TqV and TqW is a direct one, denoted by ⊕.

Definition 2.6 (see [12]). LetX be a Banach space, F : X → X be a map, and z ∈ X be a saddlepoint of F.

(i) An orbitOF(p0) = {pj}+∞j=−∞ is said to be a homoclinic orbit (asymptotic) to z if p0 /= z

and limj→+∞pj = limj→−∞pj = z.

(ii) A homoclinic orbit OF(p0) = {pj}+∞j=−∞ to z is said to be transversal if there existsan open neighborhood U of z such that p−i ∈ Wu

loc(z,U, F) and pj ∈ Wsloc(z,U, F)

for any sufficiently large integers i, j ≥ 0, and Fi+j sends a disc in Wuloc(z,U, F)

containing p−i diffeomorphically onto its image that is transversal to Wsloc(z,U, F)

at pj .

The following lemma is taken from Theorems 3.1 and 5.2, Corollary 6.1, and the resultin Section 7 of [12].

Lemma 2.7. Let F : Z → Z be a map, where Z = X × Y , and X and Y are Banach spaces.

(i) Let A and B be linear continuous maps in X and Y , respectively, with the absolute valuesof the spectrum of A less than 1 and the absolute values of the spectrum of B larger than 1,and ‖A‖, ‖B−1‖ ≤ λ0 for some constant 0 < λ0 < 1.

(ii) Assume that U is an open neighborhood of 0 in Z and f1 : U → X, f2 : U → Y areCk(k ≥ 1) maps with fi(0) = 0, Dfi(0) = 0, i = 1, 2. Further, assume that Df1, Df2 areuniformly continuous in U, and satisfies that for some constants 0 < θ < 1 − λ0 and γ > 0,‖Df1(x, y)‖, ‖Df2(x, y)‖ < θ for all (x, y) ∈ B(0, γ) ⊂ U.

(iii) Let F : U → Z be of the following form:

F(x, y

)=(Ax + f1

(x, y

), By + f2

(x, y

)), (2.4)

and have the local stable and unstable manifoldsWsloc(0, U, F) = {(x, y) | (x, y) ∈ U, y =

0} andWuloc(0, U, F) = {(x, y) | (x, y) ∈ U, x = 0}/= {0}.

(iv) Assume that O(p0) = {pi}∞i=−∞ is a homoclinic orbit of F with pi → 0 as i → ±∞, andthere exists an integerN > 0 such that p−N ∈ Wu

loc(0, U, F), pN ∈ Wsloc(0, U, F), and


(iv1) F2N sends a disc O1 ∩ Wuloc(0, U, F) centered at p−N diffeomorphically onto O2 =

F2N(O1) containing pN ;

(iv2) O2 intersects Wsloc(0, U, F) transversally at pN .

Then O(p0) is a transversal homoclinic orbit of F. Furthermore, there exists an integer k > 0 and asubset Λ in a neighborhood of O(p0) such that Fk on Λ is topologically conjugate to the full shift mapon the doubly infinite sequence of two symbols. Consequently, F is chaotic in the sense of both Li-Yorkeand Devaney, and its topological entropy h(F) ≥ log 2/k.

Note that it is not required that F is a diffeomorphism, even F may not be continuouson the whole space Z in Lemma 2.7.

3. Distribution of Maps with Transversal Homoclinic Orbits

In this section, we first consider distribution of maps with transversal homoclinic orbits ina continuous self-map space, which consists of continuous maps that transform a closed,bounded, and convex set in a Banach space into itself. At the end of this section, we discussdistribution of chaotic maps in a continuous map space, in which a map may not transformits domain into itself.

Without special illustration, we always assume that (X, ‖ · ‖X) and (Y, ‖ · ‖Y ) areBanach spaces, andDX andDY are bounded, convex, and open sets in X and Y , respectively.It is evident that D = DX × DY is a bounded, convex, and open set in Z = X × Y , wherethe norm ‖ · ‖ on Z is defined by ‖(x, y)‖ = max{‖x‖X, ‖y‖Y}, for any (x, y) ∈ Z, wherex ∈ X, y ∈ Y . Introduce the following map space:

C0

(D,D

):=

{f : D −→ D is continuous and has a fixed point in D

}. (3.1)

For any f ∈ C0(D,D), let

∥∥f

∥∥ := sup

{∥∥f(x)

∥∥ : x ∈ D

}, (3.2)

and for any f, g ∈ C0(D,D), let

d(f, g

):=

∥∥f − g∥∥. (3.3)

Then (C0(D,D), d) is a metric space. It may not be complete because a limit of a sequence ofmaps in C0(D,D) is continuous and bounded, but may not have a fixed point in D. But inthe special case that Z is finite-dimensional, (C0(D,D), d) is a complete metric space by theSchauder fixed point theorem.

In this section, we first study distribution of maps with transversal homoclinic orbitsin C0(D,D).


For convenience, by (x, y) ∈ Z denote x ∈ X and y ∈ Y , by Fix(f) denote the set ofall the fixed points of f . For (x1, y1), (x2, y2) ∈ Z with (x1, y1)/= (x2, y2), by l((x1, y1), (x2, y2))denote the straight half-line connecting (x1, y1) and (x2, y2):

l((x1, y1

),(x2, y2

)):=

{u =

(x1, y1

)+ t

((x2, y2

) − (x1, y1

)): t ≥ 0

}. (3.4)

Lemma 3.1 (see [22, Lemma 3.1]). For every map f ∈ C0(D,D) and any ε > 0, there exists a mapg ∈ C0(D,D) such that d(f, g) < ε, Fix(g) ∩ D/= ∅, and g is continuously differentiable in someneighborhood of some point x∗ ∈ Fix(g) ∩D.

Lemma 3.2. For every map f ∈ C0(D,D) and every ε > 0, there exists a map F ∈ C0(D,D) withd(f, F) < ε such that F has a transversal homoclinic orbit in D.

Proof. Fix any f ∈ C0(D,D). By Lemma 3.1, it suffices to consider the case that f has a fixedpoint z = (z1, z2) ∈ D with z1 ∈ X, z2 ∈ Y , and is continuously differentiable in some neigh-borhood of z.

For any ε > 0, there exists a positive constant r0 < ε/4 with B(z, r0) ⊂ D such that

∥∥f

(x, y

) − z∥∥ <

ε

4,

(x, y

) ∈ B(z, r0). (3.5)

Let r = r0/4, take two constants a, b with r < a < b < r0/3 and take two pointsp1 = (x0, z2) ∈ B(z, r), p0 = (z1, y0) ∈ B(z, b) \ B(z, a), where x0 ∈ X, y0 ∈ Y .

The rest of the proof is divided into three steps.

Step 1. Construct a map F that is locally controlled near z.Define

F(x, y

)=

⎧⎨

⎩

(λ(x − z1) + z1, μ

(y − z2

)+ z2

),

(x, y

) ∈ B(z, r),(x + (x0 − z1), y − (

y0 − z2)),

(x, y

) ∈ B(z, b) \ B(z, a),(3.6)

where x ∈ X, y ∈ Y , and λ and μ are real parameters and satisfy

|λ| < 1,b

r<∣∣μ∣∣ <

r0r. (3.7)

Note that |μ| > 1 since r < b.For any (x, y) ∈ B(z, a) \ B(z, r), F(x, y) is defined as follows. Let (x′

1, y′1) and

(x′2, y

′2) be the intersection points of the straight line l(z, (x, y)) with ∂B(z, r) and ∂B(z, a),

respectively, (see Figure 1). Set

F(x, y

)= F

(x′1, y

′1

)+ t

(x, y

)(F(x′2, y

′2) − F

(x′1, y

′1

)), (3.8)

where t(x, y) ∈ (0, 1) is determined as follows;

(x, y

)=(x′1, y

′1

)+ t

(x, y

)((x′2, y

′2

) − (x′1, y

′1

)). (3.9)


X

Y

O

z2 z p1

r

ab

p0(x′2, y

′2)

(x, y)

(x′1, y

′1)

z1

Figure 1

It is noted that when (x, y) continuously varies inB(z, a)\B(z, r), so do the intersection points(x′

1, y′1) and (x′

2, y′2). Consequently, t(x, y) and then F(x, y) are continuous in B(z, a)\B(z, r).

Next, define F(x, y) = f(x, y) for (x, y) ∈ D \ B(z, r0). Finally, for any (x, y) ∈B(z, r0) \ B(z, b), suppose that (x′′

1, y′′1) and (x′′

2, y′′2) are the intersection points of the straight

line l(z, (x, y)) with ∂B(z, b) and ∂B(z, r0), respectively. Define F(x, y) as that in (3.8), wheret(x, y) is determined by (3.9) with (x′

1, y′1) and (x′

2, y′2) replaced by (x′′

1, y′′1) and (x′′

2, y′′2),

respectively. Hence, F(x, y) is continuous in B(z, r0) \ B(z, b).Obviously, z is a saddle fixed point of F, and

{(x, y

): x = z1,

∥∥y − z2∥∥Y< b

} ⊂ Wuloc(z, b, F),

{(x, y

): y = z2, ‖x − z1‖X < r

} ⊂ Wsloc(z, r, F).

(3.10)

Step 2. F ∈ C0(D,D) and satisfies that d(f, F) < ε.From the definition of F, it is easy to know that F is continuous on D and has a fixed

point z ∈ D, that is, F ∈ C0(D,D).Next, we will prove that d(f, F) < ε. For (x, y) ∈ D\B(z, r0), ‖F(x, y)−f(x, y)‖ = 0 < ε.

For (x, y) ∈ B(z, r), it follows from (3.5) and (3.6) that

∥∥F(x, y

) − f(x, y

)∥∥ ≤ ∥∥F(x, y

) − z∥∥ +

∥∥f(x, y

) − z∥∥ ≤ r0 +

ε

4< ε. (3.11)

For (x, y) ∈ B(z, a) \ B(z, r), it follows from (3.5), (3.6), and (3.8) that

∥∥F(x, y

) − f(x, y

)∥∥ =∥∥F

(x′1, y

′1

)+ t

(x, y

)(F(x′2, y

′2) − F

(x′1, y

′1

)) − f(x, y

)∥∥

≤ ∣∣1 − t

(x, y

)∣∣∥∥F

(x′1, y

′1

) − z∥∥ +

∣∣t(x, y

)∣∣∥∥F

(x′2, y

′2) − z

∥∥ +

∥∥f

(x, y

) − z∥∥

≤ r0 + 2b +ε

4< ε.

(3.12)


For (x, y) ∈ B(z, b) \ B(z, a), from (3.5) and (3.6), one has

∥∥F(x, y

) − f(x, y

)∥∥ ≤ ∥∥F(x, y

) − z∥∥ +

∥∥f(x, y

) − z∥∥ < 2b +

ε

4< ε. (3.13)

For (x, y) ∈ B(z, r0) \ B(z, b), from (3.5), (3.6), and (3.8), one has

∥∥F(x, y

) − f(x, y

)∥∥ =∥∥F

(x′′1, y

′′1

)+ t

(x, y

)(F(x′′2, y

′′2) − F

(x′′1, y

′′1

)) − f(x, y

)∥∥

≤ ∣∣1 − t(x, y

)∣∣∥∥F(x′′1, y

′′1

) − z∥∥ +

∣∣t(x, y

)∣∣∥∥F(x′′2, y

′′2) − z

∥∥ +∥∥f

(x, y

) − z∥∥

≤ r0 +ε

4+ε

4< ε.

(3.14)

Therefore, from the above discussion, d(f, F) = ‖F − f‖ < ε.

Step 3. F has a transversal homoclinic orbit in D.It follows from (3.6) that F(p−1) = p0 and F(p0) = p1, where p−1 = (z1, μ−1(y0−z2)+z2) ∈

B(z, r) by (3.7). So, by (3.10) one has that p−1 ∈ Wuloc(z, r, F), p1 ∈ Ws

loc(z, r, F). Hence,O(p0) ={pj}∞j=−∞ satisfies that limj→∞pj = limj→−∞pj = z. Thus, O(p0) ⊂ D and is a homoclinic orbitof F.

Set a positive constant δ satisfying

δ < min{b − ∥

∥y0 − z2∥∥Y,∥∥y0 − z2

∥∥Y− a,

∣∣μ∣∣−1b, r − ∣

∣μ∣∣−1b

}. (3.15)

Then 0 < δ < r by (3.7), and consequently it follows from (3.10) that the disc

O0 :={(x, y

): x = z1,

∥∥y − y0

∥∥Y ≤ δ

} ⊂(B(z, b) \ B(z, a)

)∩Wu

loc(z, b, F). (3.16)

Further, set the discs

O−1 :={(

x, y): x = z1,

∥∥∥y −

(μ−1(y0 − z2

)+ z2

)∥∥∥Y≤ ∣∣μ

∣∣−1δ},

O1 :={(x, y

): x = x0,

∥∥y − z2∥∥Y ≤ δ

}.

(3.17)

Then O−1 ⊂ Wuloc(z, r, F), O0 = F(O−1), and O1 = F(O0) (see Figure 2).

It is evident thatO1 intersectsWsloc(z, r, F) transversally at point p1. In addition, by the

definition of F in B(z, b), one can get that F2 : O−1 → O1 is a diffeomorphism. Therefore,O(p0) is a transversal homoclinic orbit asymptotic to z of F by Definition 2.6, whereN = 1.

The entire proof is complete.

Theorem 3.3. Let X and Y be Banach spaces, Z = X × Y , and D be a bounded, convex, and openset in Z. Then, for every map f ∈ C0(D,D) and for any ε > 0, there exists a map F ∈ C0(D,D)satisfying


X

Y

O

Wsloc(z, r, F)

Wuloc(z, r, F)

z

O−1O1

O0

p−1

p1

p0

Figure 2

(1) d(f, F) < ε;

(2) F has a transversal homoclinic orbit in D;

(3) F is chaotic in the sense of both Li-Yorke and Devaney;

(4) the topological entropy h(F) > 0.

Proof. Let F be defined as in Lemma 3.2. Then (1) and (2) hold by Lemma 3.2.Let F, r, λ, μ, andU be specified in the proof of Lemma 3.2. Without loss of generality,

suppose that the fixed point z of F is the origin.Set U = B(0, r), A = λI, B = μI, f1|U = 0, f2|U = 0. So A and B satisfy assumption

(i) in Lemma 2.7, where λ0 = max{|λ|, |μ|−1} < 1. Take θ = (1 − λ0)/2. Then ‖Df1(x, y)‖ =‖Df2(x, y)‖ = 0 < θ < 1 − λ0 for (x, y) ∈ B(0, r). Further,

Wuloc(0, r, F) =

{(x, y

): x = 0,

∥∥y

∥∥Y< r

},

Wsloc(0, r, F) =

{(x, y

): y = 0, ‖x‖X < r

}.

(3.18)

Hence, F satisfies assumptions (ii) and (iii) in Lemma 2.7 with γ = r.By the discussions in Step 3 in the proof of Lemma 3.2, F satisfies assumption (iv) in

Lemma 2.7, where O(p0), O−1, O0, and O1 are the same as those in the proof of Lemma 3.2andN = 1. So, all the assumptions in Lemma 2.7 are satisfied. Consequently, (3) and (4) holdby Lemma 2.7. The proof is complete.

When it is not required that a map transforms its domain D into itself, the convexityof domain D can be removed and all the corresponding results to Lemmas 3.1 and 3.2 andTheorem 3.3 still hold. In detail, let S be a bounded open set in Z and

C0

(S,Z

):=

{f : S −→ Z is continuous and bounded, and has a fixed point in S

}. (3.19)

Then (C0(S,Z), d) is a metric space, where d is defined the same as that in (3.3). The resultsof Lemma 3.2 and Theorem 3.3 hold, where C0(D,D) is replaced by C0(S,Z). Their proofsare similar.

Now, we only present the detailed result corresponding to Theorem 3.3.


Theorem 3.4. Let X and Y be Banach spaces, Z = X×Y , and S be a bounded open set inZ. Then, forevery map f ∈ C0(S,Z) and for any ε > 0, there exists a map F ∈ C0(S,Z) satisfying the following

(1) d(f, F) < ε;

(2) F has a transversal homoclinic orbit in S;

(3) F is chaotic in the sense of both Li-Yorke and Devaney;

(4) the topological entropy h(F) > 0.

Remark 3.5. A general Banach space Z may not be discomposed into a product of two Banachspaces with dimension greater than or equal to 1. However, it is true for Z = Rn with n ≥ 2.So Theorem 3.3 holds for each n-dimensional space Rn with n ≥ 2. In addition, if D is abounded and convex set in Rn, every continuous map f : D → D has a fixed point in D bythe Schauder fixed point theorem. In this case one has that

C0

(D,D

)= C

(D,D

)={f : D −→ D is continuous

}. (3.20)

Remark 3.6. (1) As we all know, under C1 perturbation, the hyperbolicity of a map is pre-served. But it is obvious that the conclusion does not hold in the C0 sense.

(2) In the C0 topology, Theorems 3.3 and 3.4 show the density of distributions ofmaps with transversal homoclinic orbits, and consequently in the sense of both Li-Yorke andDevaney. However, it is not true in the C1 topology. For example, consider the map

f(x, y

)=(x2,y

2

),

(x, y

) ∈ I2 = [0, 1] × [0, 1]. (3.21)

Clearly, f ∈ C0(I2, I2) and z = 0 is a globally asymptotically stable fixed point of f in I2. ByTheorem 3.3, for each ε > 0, there exists a map F ∈ C0(I2, I2) with ‖F − f‖ < ε such that F ischaotic in the sense of both Li-Yorke and Devaney. But, in the C1 topology, for each positiveconstant ε < 1/2 and for every map F ∈ C1(I2, I2) with

∥∥F − f

∥∥C1 :=max

{∥∥F − f∥∥,

∥∥F ′ − f ′∥∥} < ε, (3.22)

F is globally asymptotically stable in I2, and so is not chaotic in any sense.

Acknowledgments

This paper was supported by the RFDP of Higher Education of China (Grant20100131110024) and the NNSF of China (Grant 11071143).

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